Self-Covering, finiteness, and fibering over a circle
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866913644890030080 |
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| author | Qin, Lizhen Su, Yang Wang, Botong |
| author_facet | Qin, Lizhen Su, Yang Wang, Botong |
| contents | A topological space is called self-covering if it is a nontrivial cover of itself. We prove that a closed self-covering manifold $M$ with free abelian fundamental group fibers over a circle under certain assumptions. In particular, we give a complete answer to the question whether a self-covering manifold with fundamental group $\mathbb Z$ is a fiber bundle over $S^1$, except for the $4$-dimensional smooth case. As an algebraic Hilfssatz, we develop a criterion for finite generation of modules over a commutative Noetherian ring. We also construct examples of self-covering manifolds with non-free abelian fundamental group, which are not fiber bundles over $S^1$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_11750 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Self-Covering, finiteness, and fibering over a circle Qin, Lizhen Su, Yang Wang, Botong Geometric Topology Commutative Algebra Algebraic Topology 57N16, 57R19, 57N65 A topological space is called self-covering if it is a nontrivial cover of itself. We prove that a closed self-covering manifold $M$ with free abelian fundamental group fibers over a circle under certain assumptions. In particular, we give a complete answer to the question whether a self-covering manifold with fundamental group $\mathbb Z$ is a fiber bundle over $S^1$, except for the $4$-dimensional smooth case. As an algebraic Hilfssatz, we develop a criterion for finite generation of modules over a commutative Noetherian ring. We also construct examples of self-covering manifolds with non-free abelian fundamental group, which are not fiber bundles over $S^1$ |
| title | Self-Covering, finiteness, and fibering over a circle |
| topic | Geometric Topology Commutative Algebra Algebraic Topology 57N16, 57R19, 57N65 |
| url | https://arxiv.org/abs/2112.11750 |